Random Walks: WEEK 5 1 Preliminary: reversible chains
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چکیده
• We can still talk about reversibility if the chain is only irreducible and positive-recurrent. • If one assumes that the chain is in stationary distribution from the start, then the backwards chain Xn, Xn−1, . . . has the same transition probabilities as the original chain, hence the name “reversible”. • If π∗ satisfies the detailed balance equation, then π∗ = π∗P . • The reciprocal statement is wrong, as we will see in some counter-examples. • Note that in general, the detailed balance equation is easier to solve than the equation π∗ = π∗P , but there are unfortunately no simple conditions that ensure that the detailed balance equation is satisfied.
منابع مشابه
Random Walks on Infinite Graphs and Groups — a Survey on Selected Topics
Contents 1. Introduction 2 2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4 C. Random walks on groups 5 D. Group-invariant random walks on graphs 6 E. Harmonic and superharmonic functions 6 3. Spectral radius, amenability and law of large numbers 6 A. Spectral radius, isoperimetric inequalities and growth 6 B. Law of large numbers 9 ...
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